Elements of Geometry and Conic Sections |
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Page 98
Elias Loomis. BOOK VI . REGULAR POLYGONS , AND THE AREA OF THE CIRCLE . Definition . A regular polygon is one which is both equiangular and equilateral . An equilateral triangle is a regular polygon of three sides ; a square is one of ...
Elias Loomis. BOOK VI . REGULAR POLYGONS , AND THE AREA OF THE CIRCLE . Definition . A regular polygon is one which is both equiangular and equilateral . An equilateral triangle is a regular polygon of three sides ; a square is one of ...
Page 99
... regular polygon are de'er- mined by the number of its sides . PROPOSITION II . THEOREM . A circle may be described about any regular polygon , and another may be inscribed within it . Let ABCDEF be any regular polygon ; a circle may be ...
... regular polygon are de'er- mined by the number of its sides . PROPOSITION II . THEOREM . A circle may be described about any regular polygon , and another may be inscribed within it . Let ABCDEF be any regular polygon ; a circle may be ...
Page 100
... regular polygon of any number of sides in a circle , it is only necessary to divide the circum- ference into the same number of equal parts ; for , if the arcs are equal , the chords AB , BC , CD , & c . , will be equal . Hence the ...
... regular polygon of any number of sides in a circle , it is only necessary to divide the circum- ference into the same number of equal parts ; for , if the arcs are equal , the chords AB , BC , CD , & c . , will be equal . Hence the ...
Page 102
... regular dec- agon , a regular pentagon may be inscribed in the circle . Cor . 2. By combining this Proposition with ... polygon , another polygon of double the number of sides may be inscribed in a circle . Hence the square will enable ...
... regular dec- agon , a regular pentagon may be inscribed in the circle . Cor . 2. By combining this Proposition with ... polygon , another polygon of double the number of sides may be inscribed in a circle . Hence the square will enable ...
Page 103
... polygon are equal to each other . Hence this polygon is regular , and similar to the one inscribed . Cor . 1. Conversely , if the circumscribed polygon is given , and it is required to form the similar inscribed one , draw the lines OL ...
... polygon are equal to each other . Hence this polygon is regular , and similar to the one inscribed . Cor . 1. Conversely , if the circumscribed polygon is given , and it is required to form the similar inscribed one , draw the lines OL ...
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Common terms and phrases
ABCD AC is equal allel altitude angle ABC angle ACB angle BAC base BCDEF bisected chord circle circumference cone convex surface curve described diagonals diameter draw ellipse equal angles equal to AC equally distant equiangular equilateral equivalent exterior angle foci four right angles frustum given angle given point greater hyperbola hypothenuse inscribed intersect join latus rectum Let ABC lines AC Loomis major axis mean proportional measured by half meet number of sides ordinate parabola parallelogram parallelopiped pendicular perimeter perpen perpendicular plane MN prism Professor of Mathematics PROPOSITION pyramid radii radius ratio rectangle regular polygon right angles Prop Scholium segment side AC similar similar triangles slant height solid angle sphere spherical triangle square subtangent tangent THEOREM triangle ABC vertex vertices
Popular passages
Page 60 - Any two rectangles are to each other as the products of their bases by their altitudes.
Page 17 - If two triangles have two sides, and the included angle of the one, equal to two sides and the included angle of the other, each to each, the two triangles will be equal, their third sides will be equal, and their other angles will be equal, each to each.
Page 101 - When you have proved that the three angles of every triangle are equal to two right angles...
Page 63 - IF a straight line be divided into any two parts, the square of the whole line is equal to the squares of the two parts, together with twice the' rectangle contained by the parts.
Page 18 - BC common to the two triangles, which is adjacent to their equal angles ; therefore their other sides shall be equal, each to each, and the third angle of the one to the third angle of the other, (26.
Page 10 - When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle ; and the straight line which stands on the other is called a perpendicular to it.
Page 32 - All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.
Page 37 - Proportional, when the ratio of the first to the second is equal to the ratio of the second to the third.
Page 15 - Wherefore, when a straight line, &c. QED PROP. XIV. THEOR. If, at a point in a straight line, two other straight lines, upon the opposite sides of it, make the adjacent angles together equal to two right angles, these two straight lines shall be in one and the same straight line.
Page 44 - A Circle is a plane figure bounded by a curved line every point of which is equally distant from a point within called the center.